Animating Dynamic Paint with noise texture
It sounds as if you've done most of this already.. but will illustrate anyway..
The Canvas
A plane, subdivided sufficiently for resulting texture resolution.

Format: vertex colors, as per your example, no effects (Drying, etc)
We're only interested in the Paintmap


The Brush

Color, White, this is just going to be a mask, Alpha: 1, this example isn't overlaying anything.
Wetness: 1 -  the full 'load' of paint should be available immediately
Proximity will do, we can't see inside the rocks
A Color ramp for falloff. A nice soft falloff will expose the waviness of the noise in the material, gradually decreasing away from the rock. Also provides another adjustment for the final effect.


The Brush, on its own:

The Nodes
Here, the Brush color is used as a multiplier on an animated Noise texture, before the product is put through a threshold in a Color Ramp, as part of the river's material.

If you've UV mapped your river, you may want to use that texture coordinate.
The Y location in the Mapping node is keyframed to make the Noise 'flow' down the river
An Attribute node is used to pick up the vertex color layer. Just enter its name in the text field.
Strictly, I probably should have used the 'Fac' output of the noise node for a 0 to 1 output, but clamping the multiply and tweaking elsewhere seems to be enough


The result - tweak the brush Falloff, texture scale, threshold in the Color Ramp, to vary the effect.